Session 07-05 - Bayes’ Theorem

Section 07: Probability & Statistics

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Session 07-04

If \(P(A) = 0.5\), \(P(B) = 0.4\), and \(P(A \cap B) = 0.2\), find \(P(A|B)\).
A bag has 4 red and 6 blue balls. Two are drawn without replacement. Find \(P(\text{both blue})\).
Given \(P(B|A) = 0.6\) and \(P(A) = 0.3\), find \(P(A \cap B)\).
If \(P(A|B) = P(A)\), what can we conclude about A and B?

Learning Objectives

What You’ll Master Today

Apply Bayes’ Theorem: \(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\)
Understand prior and posterior probabilities
Calculate sensitivity and specificity for diagnostic tests
Compute PPV and NPV (positive/negative predictive values)
Solve medical testing problems - a key exam topic!

Bayes’ Theorem appears on virtually every Feststellungsprüfung!

Part A: Bayes’ Theorem

Reversing Conditional Probabilities

The problem: We often know \(P(B|A)\) but need \(P(A|B)\).

Example: - We know \(P(\text{positive test}|\text{disease})\) (sensitivity) - We need \(P(\text{disease}|\text{positive test})\) (PPV)

These are not the same! This is a common misconception.

Bayes’ Theorem Formula

Bayes’ Theorem (Satz von Bayes)

\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]

Using the law of total probability:

\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|A') \cdot P(A')}\]

Understanding the Components

Term	Name	Meaning
\(P(A)\)	Prior	Initial probability before evidence
\(P(A\\|B)\)	Posterior	Updated probability after evidence
\(P(B\\|A)\)	Likelihood	How likely is evidence given A?
\(P(B)\)	Evidence	Total probability of evidence

\[\text{Posterior} = \frac{\text{Likelihood} \times \text{Prior}}{\text{Evidence}}\]

Part B: Medical Testing Framework

Key Terminology

Medical Test Metrics

Metric	Formula	Meaning
Sensitivity	\(P(+\\|D)\)	Correctly identifies sick people
Specificity	\(P(-\\|D')\)	Correctly identifies healthy people
Prevalence	\(P(D)\)	Proportion with disease in population
PPV	\(P(D\\|+)\)	Probability of disease given positive test
NPV	\(P(D'\\|-)\)	Probability of no disease given negative test

The 2×2 Table

	Disease (+)	No Disease (−)	Total
Test +	True Positive (TP)	False Positive (FP)	Test +
Test −	False Negative (FN)	True Negative (TN)	Test −
Total	Disease	No Disease	Population

Sensitivity = \(\frac{TP}{TP + FN}\)
Specificity = \(\frac{TN}{TN + FP}\)
PPV = \(\frac{TP}{TP + FP}\)
NPV = \(\frac{TN}{TN + FN}\)

Example: COVID Test

A rapid COVID test has:

Sensitivity: 95% (correctly identifies 95% of infected people)
Specificity: 98% (correctly identifies 98% of healthy people)
Prevalence: 2% (2% of population currently infected)

Question: If you test positive, what’s the probability you actually have COVID?

This is asking for PPV = \(P(D|+)\)!

Solution Using Bayes’ Theorem

\[P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+)}\]

Calculate \(P(+)\) using law of total probability: \[P(+) = P(+|D) \cdot P(D) + P(+|D') \cdot P(D')\] \[= 0.95 \times 0.02 + 0.02 \times 0.98 = 0.019 + 0.0196 = 0.0386\]

Apply Bayes: \[P(D|+) = \frac{0.95 \times 0.02}{0.0386} = \frac{0.019}{0.0386} \approx 0.492\]

Only about 49% of positive tests are true positives when prevalence is low!

Visual: Why PPV Can Be Low

The Prevalence Effect

PPV depends heavily on prevalence!

Break - 10 Minutes

Part C: Systematic Problem-Solving

Step-by-Step Approach

Strategy for Bayes Problems

Identify what you need: Usually \(P(D|+)\) or \(P(D|-)\)
Extract given information: Sensitivity, specificity, prevalence
Set up the formula: Write Bayes’ theorem
Calculate \(P(+)\) or \(P(-)\): Use law of total probability
Substitute and solve: Careful with arithmetic!
Interpret: What does the answer mean?

Complete Example: Disease Screening

A screening test for a disease has:

Sensitivity = 90%
Specificity = 95%
Prevalence = 1%

Find: a) PPV b) NPV

Solution Part a) PPV

\[P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+|D) \cdot P(D) + P(+|D') \cdot P(D')}\]

Given values: - \(P(+|D) = 0.90\) (sensitivity) - \(P(D) = 0.01\) (prevalence) - \(P(+|D') = 1 - 0.95 = 0.05\) (false positive rate) - \(P(D') = 0.99\)

\[P(D|+) = \frac{0.90 \times 0.01}{0.90 \times 0.01 + 0.05 \times 0.99}\] \[= \frac{0.009}{0.009 + 0.0495} = \frac{0.009}{0.0585} \approx 0.154\]

Solution Part b) NPV

\[P(D'|-) = \frac{P(-|D') \cdot P(D')}{P(-)}\]

Calculate \(P(-)\): \[P(-) = P(-|D) \cdot P(D) + P(-|D') \cdot P(D')\] \[= 0.10 \times 0.01 + 0.95 \times 0.99 = 0.001 + 0.9405 = 0.9415\]

\[P(D'|-) = \frac{0.95 \times 0.99}{0.9415} = \frac{0.9405}{0.9415} \approx 0.999\]

PPV is only 15.4%, but NPV is 99.9%! A negative result is very reliable.

Part D: Contingency Table Method

Alternative Approach

Use a hypothetical population (e.g., 10,000 people):

	Disease	No Disease	Total
Test +
Test −
Total	100	9,900	10,000

Fill in using sensitivity and specificity:

	Disease	No Disease	Total
Test +	90	495	585
Test −	10	9,405	9,415
Total	100	9,900	10,000

Read directly: PPV = \(\frac{90}{585} = 0.154\), NPV = \(\frac{9405}{9415} = 0.999\)

Guided Practice - 20 Minutes

Practice Problem 1

A factory has two machines:

Machine A produces 60% of items, with 3% defect rate
Machine B produces 40% of items, with 5% defect rate

If a randomly selected item is defective, what’s the probability it came from Machine A?

Practice Problem 2 (Exam-Style)

A medical test has sensitivity 85% and specificity 90%.

In a population with 5% prevalence:

Calculate PPV
Calculate NPV
Construct a contingency table for 1000 people
Interpret your results

Wrap-Up & Key Takeaways

Today’s Essential Concepts

Bayes’ Theorem: \(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\)
Medical testing: Sensitivity, specificity, prevalence
PPV and NPV: What positive/negative results mean
Prevalence matters: Low prevalence → Low PPV
Two methods: Formula or contingency table

Next Session Preview

Coming Up: Contingency Tables

Constructing tables from word problems
Reading marginal, joint, and conditional probabilities
Independence testing in tables
Exam-style problems

Homework

Complete Tasks 07-05 - especially the medical testing problems!